The
QZ Algorithm for the Calculation of the Eigenvalues of a Real MatrixD. J. Evans and W. S. Yousif
Parallel Algorithms and Applications, Vol. 4, pp. 183-192, 1994
Abstract
In this paper we present a new algorithm for calculating the eigenvalues of a real matrix. The algorithm is based on the orthogonal decomposition of a square dense matrix by the QZ method proposed in the paper “The QZ orthogonal decomposition method” . A comparison with the QR algorithm confirm the new algorithm to be computationally superior.
A new orthogonal factorization for a square dense matrix was proposed . It decomposes a matrix A into factors of the form
where Q is an orthogonal matrix and Z has the following matrix representation
The new factorization is analogous to the Givens QR algorithm .
The QR algorithm starts with
where
Thus the sequence of matrices
The transformation matrices used for the decomposition of the
The
The repetition of the transformation yields a sequence of matrices
Hence, the eigenvalues of this Z-type matrix can be calculated in parallel as they are the eigenvalues of the